## Loyola College M.Sc. Mathematics April 2012 Complex Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – APRIL 2012

# MT 3811 – COMPLEX ANALYSIS

Date : 24-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

1. a) Prove that  if using Leibniz’s rule

OR

1. b) Let be a non-constant polynomial. Prove that there is a complex number  such that.                                                                                                       (5)
2. c) Let be an analytic function. Prove that  for  where  Hence prove that if f is analytic in an open disk  then prove that  for  where.

OR

1. d) State and prove homotopic version of Cauchy’s theorem.      (15)

1. a) State and prove Morera’s theorem.

OR

1. b) Prove that a differentiable function  on  is convex if and only if  is

increasing.                                                                                                             (5)

1. c) State and prove the Arzela-Ascoli theorem.

OR

1. d) State and prove the Riemann mapping theorem.

(15)

1. a) Show that  in the usual notation.

OR

1. b) If and  then prove that .

(5)

1. c) (i) Let be a compact metric space and let  be a sequence of continuous functions from X into  such that  converges absolutely and uniformly for x in X. Then prove that the product  converges absolutely and uniformly for x in X. Also prove that there is an integer  such that  if and only if  for some n, .

(ii) State and prove Weierstrass factorization theorem.                                     (7+8)

OR

1. d) Let , then prove that converges absolutely if and only if  converges absolutely.
2. e) State and prove Bohr-Mollerup theorem.    (7+8)

1. a) State and prove Jensen’s formula.

OR

1. b) If  is a metric space, then prove that  is also a metric on                                                                                                                                            (5)
2. c) State and prove Rung’s theorem.                                                                        (15)

OR

1. d) State and prove Hadamard’s factorization theorem.                                           (15)

1. a) Prove that any two bases of a same module are connected by a unimodular transformation.

OR

1. b) Prove that an elliptic function without poles is a constant.

(5)

1. c) (i) Prove that the zeros  and poles  of an elliptic function satisfy .

(ii) Derive Legendre’s relation                                                                           (7+8)

OR

1. d) (i) State and prove the addition theorem for the Weierstrass

(ii) Show that

(8+7)

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## Loyola College M.Sc. Mathematics Nov 2012 Complex Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – MATHEMATICS

THIRD SEMESTER – NOVEMBER 2012

# MT 3811 – COMPLEX ANALYSIS

Date : 03/11/2012            Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

1. a) Prove that  if  using Leibniz’s rule.

OR

1. b) State and prove Liouville’s theorem.        (5)
2. c) State and prove first version of Cauchy’s integral formula.

OR

1. d) State and prove the homotopic version of Cauchy’s theorem                  (15)

1. a) State and prove Hadamard’s three circles theorem.

OR

1. b) Define a convex function and prove that a function  is convex if and

only if the set  is a convex set.                           (5)

1. c) State and prove Goursat’s theorem.

OR

1. d) State and prove Arzela Ascoli theorem.      (15)

1. a) Let , for all . Then prove that  converges to a complex number different from zero if and only if  converges.

OR

1. b) Show that in the usual notation.                                                                                                                                            (5)

1. c) (i) If and  then prove that .

(ii) Prove that .

(iii) State and prove Gauss’s Formula.                                                                                                                                                                                              (5+5+5)

OR

1. d) (i) State and prove Bohr-Mollerup theorem.

(ii) Prove that (a)  converges to  in  and (b) if  then  for all .                                                                                                                                                                                          (8+7)

1. a) State and prove Jensen’s formula.

OR

1. b) Let  be a rectifiable curve and let K be a compact set such that .  If f is a continuous function on  and  then prove that there is a rational function  having all its poles on  and  such that  for all z in K.

(5)

1. c) State and prove Mittag-Leffler’s theorem.                                                          (15)

OR

1. d) State and prove Hadamard’s Factorization theorem.

(15)

1. a) Prove that any two bases of a same module are connected by a unimodular transformation.

OR

1. b) Show that and it is an odd function.                                                                                                                                                        (5)

1. c) (i) Prove that the zeros and poles  of an elliptic function satisfy .

(ii) Prove that   .                                  (7+8)

OR

1. d) (i) Show that

(ii) State and prove the addition theorem for the Weierstrass -function.                                                                                                                                                 (7+8)

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## Loyola College B.Sc. Mathematics April 2008 Complex Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

# XZ 22

SIXTH SEMESTER – APRIL 2008

# MT 6600 – COMPLEX ANALYSIS

Date : 16/04/2008             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

Answer ALL questions:                                                                    (10 x 2 = 20)

1. Prove that for any two complex numbers and ,

.

1. Verify Canchy – Riemann equations for the function
1. Show that is harmonic.
2. Define Möbius transformation.
3. State Cauchy – Goursat theorem.
4. State Cauchy’s Integral formula.
5. Evaluate where C is the circle  .
6. Find the Taylor’s series expansion about z=0.
7. Obtain the Laurent’s series for  in .
8. Find the residue of at .

SECTION – B

Answer any  FIVE questions:                                                          (5 x 8 = 40)

1. Prove that the function

is not differentiable at the origin, but Canchy-Riemann equations are satisfied there.

1. Show that the function is harmonic and find the corresponding analytic function.
2. Show that the transformation maps the circle onto a straight line .
3. Evaluate  if C is the positively oriented circle
4. State and prove Liouville’s theorem.
5. State and prove Residue theorem.
6. State and prove Taylor’s theorem.
7. Find the residues of  at its poles.

SECTION – C

Answer any  TWO questions:                                                           (2 x 20 = 40)

1. a) If  is a regular function of , prove that
1. b) Find the bilinear transformation that maps the points onto the points .
1. State and prove Cauchy’s theorem.
2. Expand in the regions.
1. i)

ii)

iii)

1. Using the method of Contour integration, prove that

i)

ii)

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## Loyola College B.Sc. Mathematics April 2009 Complex Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

 ZA 36

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – April 2009

# MT 6603/MT 6600 – COMPLEX ANALYSIS

Date & Time: 18/04/2009 / 9:00 – 12:00        Dept. No.                                                          Max. : 100 Marks

PART – A

Answer ALL questions:                                                                               (10 x 2 = 20)

1. Find the absolute value of .
2. Show that does not have a limit as .
3. When do you say that a sequence of function converges uniformly?
4. Using Cauchy integral formula evaluate where C is the circle .
5. Find zeros of the function .
6. When do you say that a point “a” is an isolated singularity of ?
7. Calculate the residue of .
8. State Cauchy’s residue theorem.
9. Define a bilinear transformation.
10. When do you say that a function is conformal at ?

PART – B

Answer any FIVE questions:                                                                                   (5 x 8 = 40)

1. Illustrate by an example that C.R. equations are not sufficient for differentiability at a point.
2. State Liouville’s theorem and deduce the Fundamental theorem of algebra from it.
3. Let be analylic in a region D bounded by two concentric circles and on the boundary. Let be any point in D. Show that

.

1. State and prove Maximum modules theorem.
2. Expand as a power series in the region .
3. Evaluate where C is the square having vertices

.

1. Using contour integration evaluate .
2. Find the image of the circle under the map .

PART – C

Answer any TWO questions:                                                                       (2 x 20 = 40)

1. (a) Let nbe a function defined on the region D such that u and v and

their first order partial derivatives are continuous in D. If the first order partial derivatives of u and v satisfy the C.R. equations at , show that  is differentiable at

(b) Let be a given power series. Show that there exists a number R such that

such that

(i)  the series converges absolutely for every z with

(ii) if , the convergence uniform in .

(iii) if , the series diverges.                                                                        (10+10)

1. (a) State and prove Taylor’s theorem.

(b) Expand using Taylors series about the point .                                       (15+4)

1. (a) State and prove Argument theorem.

(b) State and prove Rouche’s theorem.

(c) Prove that .                                                                                      (7+6+7)

1. (a) Show that the transformation maps the circle into a straight

line given by .

(b) Find the bilinear transformation which maps the points respectively to

.

(c) Show that a bilinear transformation where maps the real axis into

itself if and only if a, b, c, d are real.                                                                          (6+6+8)

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## Loyola College B.Sc. Mathematics April 2011 Complex Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2011

# MT 6603/MT 6600 – COMPLEX ANALYSIS

Date : 05-04-2011              Dept. No.                                                 Max. : 100 Marks

Time : 9:00 – 12:00

PART – A

Answer ALL Questions                                                                                          (10 x 2 = 20 marks)

1. Express the function f(z) = z3+z+1 in the form f(z) = u(x, y) + iv(x, y).
2. Show that the radius of convergence of the series
3. Find the modulus of
4. Define removable singularity and pole for a function.
5. State Morera’s theorem.
6. State Cauchy’s Residue theorem.
7. Find the points where the mapping is conformal.
8. Calculate the residues of at z = 1, 2 and 3.
9. Find the Singular points and its nature for the function e1/z.
10. Find the fixed points of the transformation w = .

PART – B

Answer any FIVE questions                                                                                  (5 x 8 = 40 marks)

1. If z1 and z2 are two complex numbers, show that
2. Given v(x,y) = find f(z) = u(x,y) + iv(x,y) such that f(z) is analytic.
3. State and Prove the fundamental theorem of algebra.
4. Obtain the Taylor’s or Laurent’s series for the function f(z) = when (i)

(ii) 1< .

1. Evaluate using Cauchy Residue theorem where C is the circle
2. State Cauchy’s theorem and Cauchy’s integral formula. Evaluate where C is positively oriented circle .
3. State and prove maximum modulus theorem.
4. Prove that the cross ratio is invariant under Mobius transformation.

PART – C

Answer any TWO questions                                                                                 (2 x 20 = 40 marks)

1. a) Prove that for the function the Cauchy – Reimann equations

are satisfied at z = 0, but f(z) is not differentiable at z = 0.

1. b) State and prove Cauchy – Hadamard’s theorem for radius of convergence. (10 + 10)

1. a) If f(z) is analytic inside and on a simple closed curve C except for a finite number of

poles inside C and has no zero on C, Prove that  where N is the

number of zeros and P is the number of poles of inside C.

1. b) Using contour integration evaluate .                                              (10 + 10)
2. a) State and Prove Taylor’s theorem.
3. b) Show that when                                   (14+6)
4. a) Let f be analytic in a region D and for Prove that  is conformal at .
5. b) Show that by means of the inversion the circle is mapped into the circle

1. c) Find the general bilinear transformation which maps the unit circle onto and

the points                                                               (10+5+5)

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## Loyola College B.Sc. Mathematics April 2012 Complex Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – APRIL 2012

# MT 6603/MT 6600 – COMPLEX ANALYSIS

Date : 16-04-2012                   Dept. No.                                          Max. : 100 Marks

Time : 1:00 – 4:00

PART – A

1. Find the absolute value of
2. Define harmonic function.
3. Find the radius of convergence of the series .
4. Using Cauchy integral formulas evaluatewhere  is the unit circle .
5. Define zero and poles of a function.
6. Write Maclaurin series expansion of s.
7. Define residue of a function.
8. State Argument principle.
9. Define isogonal mapping.
10. Define critical point.

PART – B

1. Let f(z)= .  Show that f(z) satisfies CR equations at zero but not differential at .
2. Prove that is harmonic and find its Harmonic conjugate.
3. State and prove Liouvilles theorem and deduce Fundamental theorem of algebra.
4. Find the Taylors series to represent in .
5. Suppose is analytical in the region  and is not identically zero in .Show that the set of all zeros of  is isolated.
6. Use residue calculus to evaluate dz over  where  is the unit circle.
7. Show that any bilinear transformation can be expressed as a product of translation, rotation, magnification or contraction and inversion.
8. Find the bilinear transformation which maps the points  onto

PART – C

1. (a) Derive CR equations in polar coordinates.

(b)   Find the real part of the analytic function whose imaginary part isand construct the analytic function.

20 .       (a) State and prove Cauchy integral formula.

(b)   Let F be an analytic inside and on a simple closed curve C. Let z be a point inside C. Show that f’(z) = dt.

1. (a) Expand in a Laurants series in (i) , ( ii)  .

(b) Using method of contour integration evaluate   dx.

1. (a) Show that any bilinear transformation which maps the unit circle = 1 onto  = 1 can be written in the form  where is real.

(b)State and prove Rouche’s theorem.

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## Loyola College B.Sc. Mathematics Nov 2012 Complex Analysis Question Paper PDF Download

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SIXTH SEMESTER – NOVEMBER 2012

# MT 6603/6600 – COMPLEX ANALYSIS

Date : 05/11/2012               Dept. No.                                          Max. : 100 Marks

Time : 1:00 – 4:00

PART-A

1. Show that the function is nowhere differentiable.
2. When do we say that a function is harmonic.
3. Find the radius of convergence of the series .
4. State Cauchy Goursat theorem.
5. Expand as a Taylor’s series about the point .
6. Define meromorphic function with an example.
7. Define residue of a function at a point.
8. State argument principle.
9. Define the cross ratio of a bilinear transformation.
10. Define an isogonal mapping.

PART-B

1. Show that the function is discontinuous at  given that when and .
2. Find the analytic function of which the real part is .
3. Evaluate along the closed curve containing paths and .
4. State and prove Morera’s theorem.
5. State and prove Maxmimum modulus principle.
6. Find out the zeros and discuss the nature of the singularity of .

1. State and prove Rouche’s theorem.
2. Find the bilinear transformation which maps the points into the points

PART C

1. (a) Let be a function defined in a region such that  and their first order partial derivatives are continuous in . If the first order partial derivatives of  satisfy the Cauchy-Riemann equations at a point  in D then show that f is differentiable at .

(b) Prove that every power series represents an analytic function inside its circle of convergence.

1. (a) State and prove Cauchy’s integral formula.

(b)          Expand in a Laurent’s series for (i) (ii)
(iii) .

1. (a) State and prove Residue theorem.

(b) Using contour integration evaluate  .

1. (a) Let be analytic in a region  and  for .Prove that f is conformal at .

(b) Find the bilinear transformation which maps the unit circle onto the unit circle .

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